theorem Th7:
  B is associative implies product(B,n) is associative
proof
  assume
A1: B is associative;
  now
    let x,y,z be Element of n-tuples_on D;
    thus product(B,n).(product(B,n).(x,y),z) = product(B,n).(B.:(x,y),z)
      by Def1
      .= B.:(B.:(x,y),z) by Def1
      .= B.:(x,B.:(y,z)) by A1,FINSEQOP:28
      .= product(B,n).(x,B.:(y,z)) by Def1
      .= product(B,n).(x,product(B,n).(y,z)) by Def1;
  end;
  hence thesis;
end;
